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66 lines
1.6 KiB
Plaintext
66 lines
1.6 KiB
Plaintext
Bound states of a central potential
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For any central potential V(r) = V(│r│) the eigenfunctions of H can be separated as
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❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,ϕ)
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The radial S.E. is
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⎡−͟ħ͟² ⎛ ∂͟²͟ + 2͟∂͟ ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r)
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⎣2m ⎝ ∂r² r∂r ⎠ 2 m r² ⎦
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Rₑ﹐ₗ(r) = U͟ₑ͟﹐͟ₗ͟(r)
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r
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(pic) ...
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(pic) Developed radial schrodinger equation using U(r) replacement
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- Developed normalization condition
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If V(r) is not more singular at the origin than 1/r^2 then the SE has power series solutions.
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Thus for small r we take U(r) → rˢ
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(pic) substitute U(r) = rˢ into S.E.
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-ħ²/2m [(s(s-1) + l(l+1)] + V r² = E r²
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For r → 0
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r² → 0
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V(r) r² → 0
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⇒ s(s-1) + l(l+1) = 0
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⇒ s = l+1 or s = -l
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If s = -l, the normalization conditions
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∞ │∞
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∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │ → diverges
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0 │0
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Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹
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Rₑ﹐ₗ(r) → (r→0) → rˡ
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The Hydrogen Atom
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━━━━━━━━━━━━━━━━━
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V(r) = -e²/r
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For a hydrogenic ion with nuclear charge Z
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V(r) = -Ze²/r
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Eigenfunctions:
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Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ)
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